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Register a new userHilbert transforms along variable planar curves: Lipschitz regularity
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In this paper, for $1<p<infty$, we obtain the $L^p$-boundedness of the Hilbert transform $H^{gamma}$ along a variable plane curve $(t,u(x_1, x_2)gamma(t))$, where $u$ is a Lipschitz function with small Lipschitz norm, and $gamma$ is a general curve satisfying some suitable smoothness and curvature conditions.
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The Hilbert transforms associated with monomial curves have a natural non-isotropic structure. We study the commutator of such Hilbert transforms and a symbol $b$ and prove the upper bound of this commutator when $b$ is in the corresponding non-isotropic BMO space by using the Cauchy integral trick. We also consider the lower bound of this commutator by introducing a new testing BMO space associated with the given monomial curve, which shows that the classical non-isotropic BMO space is contained in the testing BMO space. We also show that the non-zero curvature of such monomial curves are important, since when considering Hilbert transforms associated with lines, the parallel version of non-isotropic BMO space and testing BMO space have overlaps but do not have containment.
We establish an L^2 times L^2 to L^1 estimate for the bilinear Hilbert transform along a curve defined by a monomial. Our proof is closely related to multi-linear oscillatory integrals.
In this paper, we determine the $L^p(mathbb{R})times L^q(mathbb{R})rightarrow L^r(mathbb{R})$ boundedness of the bilinear Hilbert transform $H_{gamma}(f,g)$ along a convex curve $gamma$ $$H_{gamma}(f,g)(x):=mathrm{p.,v.}int_{-infty}^{infty}f(x-t)g(x-gamma(t)) ,frac{textrm{d}t}{t},$$ where $p$, $q$, and $r$ satisfy $frac{1}{p}+frac{1}{q}=frac{1}{r}$, and $r>frac{1}{2}$, $p>1$, and $q>1$. Moreover, the same $L^p(mathbb{R})times L^q(mathbb{R})rightarrow L^r(mathbb{R})$ boundedness property holds for the corresponding (sub)bilinear maximal function $M_{gamma}(f,g)$ along a convex curve $gamma$ $$M_{gamma}(f,g)(x):=sup_{varepsilon>0}frac{1}{2varepsilon}int_{-varepsilon}^{varepsilon}|f(x-t)g(x-gamma(t))| ,textrm{d}t.$$
In this paper, for general plane curves $gamma$ satisfying some suitable smoothness and curvature conditions, we obtain the single annulus $L^p(mathbb{R}^2)$-boundedness of the Hilbert transforms $H^infty_{U,gamma}$ along the variable plane curves $(t,U(x_1, x_2)gamma(t))$ and the $L^p(mathbb{R}^2)$-boundedness of the corresponding maximal functions $M^infty_{U,gamma}$, where $p>2$ and $U$ is a measurable function. The range on $p$ is sharp. Furthermore, for $1<pleq 2$, under the additional conditions that $U$ is Lipschitz and making a $varepsilon_0$-truncation with $gamma(2 varepsilon_0)leq 1/4|U|_{textrm{Lip}}$, we also obtain similar boundedness for these two operators $H^{varepsilon_0}_{U,gamma}$ and $M^{varepsilon_0}_{U,gamma}$.
Let $p(cdot): mathbb R^nto(0,infty)$ be a variable exponent function satisfying that there exists a constant $p_0in(0,p_-)$, where $p_-:=mathop{mathrm {ess,inf}}_{xin mathbb R^n}p(x)$, such that the Hardy-Littlewood maximal operator is bounded on the variable exponent Lebesgue space $L^{p(cdot)/p_0}(mathbb R^n)$. In this article, via investigating relations between boundary valued of harmonic functions on the upper half space and elements of variable exponent Hardy spaces $H^{p(cdot)}(mathbb R^n)$ introduced by E. Nakai and Y. Sawano and, independently, by D. Cruz-Uribe and L.-A. D. Wang, the authors characterize $H^{p(cdot)}(mathbb R^n)$ via the first order Riesz transforms when $p_-in (frac{n-1}n,infty)$, and via compositions of all the first order Riesz transforms when $p_-in(0,frac{n-1}n)$.
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